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I've determined that, during operation, the LHC accelerates 2808 bunches of protons with roughly 120 billion protons per bunch up to 99.9999991% the speed of light. I'd like to calculate how much total centrifugal force these protons generate per meter of the particle accelerator ring. (I recall hearing, long ago, during a tour of CERN that the bunches of protons in the beam have as much kinetic energy as a freight train). My first inclination was to calculate the mass of a proton traveling at this speed and then use mass x velocity^2 / r. However, I suspect that might be the wrong formula to use at relativistic speeds.

What is the right way to do this calculation?

Note: I read the answers to this question from 2012 but while the answers provided some equations, they did not define the variables used in those equations; therefore, those equations/answers do not answer my question.

phil1008
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The equation of motion of a particle with mass $m$ and charge $e$ moving in a pure magnetic field is given by $$\frac{d}{dt} \frac{m \vec{v}}{\sqrt{1-\vec{v}^2/c^2}} \tag{1} \label{1}= e \, \vec{v}\times \vec{B} . $$ As the energy of the particle $$\mathcal{E}= \frac{m c^2}{\sqrt{1-\vec{v}^2/c^2}}$$ does not change with time because of $\vec{v} \cdot(\vec{v} \times \vec{B})=0$, eq. \eqref{1} simplifies to $$ \frac{\mathcal{E}}{c^2} \frac{d\vec{v}}{dt}= e\, \vec{v} \times \vec{B}. \tag{2} \label{2}$$ Specializing further to the case where the particle is moving on a circle with radius $r$, eq. \eqref{2} becomes $$-\frac{\mathcal{E}}{c^2} \frac{\vec{v}^2}{r} \, \vec{e}_r = e \, \vec{v} \times \vec{B},\tag{3} \label{3}$$ with the unit vector $\vec{e}_r=\vec{r}/r$. Thus the magnitude of the force acting on the particle can be expressed in terms of its energy and the radius of the circle by the formula $$|\vec{F}|=\frac{\mathcal{E}}{r} \frac{\vec{v}^2}{c^2}, \tag{4}$$ simplifying further to $$ |\vec{F}|\simeq \frac{\mathcal{E}}{r} \tag{5}$$ in the ultrarelativistic limit $|\vec{v}|\to c$.

Inserting $r=4.25 \, {\rm km}$ for the radius of the LHC collider, $\mathcal{E} = 3 \, {\rm TeV}$ for the energy of a (single) proton and using the conversion factor ${\rm eV} = 1.6 \times 10^{-19} {\rm J}$, it is now a simple task to finish your exercise.

Hyperon
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